Let \(K\)  denote the fraction field of \(R\)  and let \(\pi \colon Y\rightarrow X\)  be a finite étale (surjective) morphism of schemes over \(R\)  with \(Y\)  integral. Then the base change \(\pi _K \colon Y_K \rightarrow X_K\)  of \(\pi \)  along \(\operatorname{Spec}K \rightarrow \operatorname{Spec}R\)  is also finite étale surjective. By assumption, the set \(X(R)^{(1)}\)  is not thin in \(X_K\) .
We follow the proof of the Chevalley–Weil theorem given in [1]} (see also [2]}) to show that there exists a finite field extension \(K^{\prime }/K\)  such that \(X(R)^{(1)} \subseteq \pi (Y(K^{\prime }))\) .
For each \(x\in X(R)^{(1)}\) , choose a dense open \(U_x\subseteq \operatorname{Spec}R\)  with complement of codimension at least two such that \(x\)  defines a morphism \(x \colon U_x \rightarrow X\) . Set
\(V_x := U_x \times _X Y\) , so that \(V_x \rightarrow U_x\)  is finite étale. Since \(R\)  is regular, by Zariski–Nagata purity of the branch locus [3]}, this morphism extends to a finite étale morphism
\(\operatorname{Spec}S_x \rightarrow \operatorname{Spec}R\) 
of degree \(\deg \pi \) .
(Here \(S_x\)  is the normalization of \(R\)  in \(V_x\) .)
By Hermite's finiteness theorem [4]}, there exist only finitely many finite étale coverings of \(\operatorname{Spec}R\)  of given degree.
Therefore, there exists a finite field extension \(K^{\prime }\)  of \(K\)  such that \(X(R)^{(1)} \subseteq \pi (Y(K^{\prime }))\) .
Let \(X^{\prime } \subseteq X_K\)  be a dense open affine. By the finiteness of \(\pi _K\) , the preimage \(Y^{\prime } = \pi _K^{-1}(X^{\prime })\)  is also affine.
In particular, the restriction \(Y^{\prime } \rightarrow X^{\prime }\)  of \(\pi _K\)  is a finite surjective morphism of quasi-projective integral varieties over \(K\) . Set \(T := \pi _K^{-1}(X(R)^{(1)}) \cap Y^{\prime }(K^{\prime }) \subseteq Y^{\prime }(K^{\prime })\) , and note that
\(\pi _K(T) = X(R)^{(1)} \cap X^{\prime } \subseteq X^{\prime }(K).\) 
Assume for a contradiction that \(\deg \pi > 1\) . Then, by Lemma REF , we have
finitely many quasi-projective integral varieties \(Y_i\)  over \(K\) 
and dominant rational maps \(\pi _i \colon Y_i \rightarrow X^{\prime }\) , each of degree \(>1\) , such that
\(\pi _K(T ) \subseteq \bigcup _i \pi _i(Y_i(K))\) .
From this it follows that
\(X(R)^{(1)} \setminus \bigcup _i \pi _i(Y_i(K)) \subseteq X_K\setminus X^{\prime }\) 
is not dense in \(X_K\) . This contradicts the fact that \(X\)  has the Hilbert property over \(R\) .
Thus, \(\deg \pi =1\) , as required.
